Global Existence for a Strongly Coupled Cahn-Hilliard System with Viscosity

Pierluigi Colli; Gianni Gilardi; Paolo Podio-Guidugli; Jürgen Sprekels

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 3, page 495-513
  • ISSN: 0392-4041

Abstract

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An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [3]. Both systems conform to the general theory developed in [5]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter ρ and the chemical potential μ . In the system studied in this note, a phase-field equation in ρ fairly more general than in [3] is coupled with a highly nonlinear diffusion equation for μ , in which the diffusivity coefficient is allowed to depend nonlinearly on both variables.

How to cite

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Colli, Pierluigi, et al. "Global Existence for a Strongly Coupled Cahn-Hilliard System with Viscosity." Bollettino dell'Unione Matematica Italiana 5.3 (2012): 495-513. <http://eudml.org/doc/290927>.

@article{Colli2012,
abstract = {An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [3]. Both systems conform to the general theory developed in [5]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $\rho$ and the chemical potential $\mu$. In the system studied in this note, a phase-field equation in $\rho$ fairly more general than in [3] is coupled with a highly nonlinear diffusion equation for $\mu$, in which the diffusivity coefficient is allowed to depend nonlinearly on both variables.},
author = {Colli, Pierluigi, Gilardi, Gianni, Podio-Guidugli, Paolo, Sprekels, Jürgen},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {495-513},
publisher = {Unione Matematica Italiana},
title = {Global Existence for a Strongly Coupled Cahn-Hilliard System with Viscosity},
url = {http://eudml.org/doc/290927},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Colli, Pierluigi
AU - Gilardi, Gianni
AU - Podio-Guidugli, Paolo
AU - Sprekels, Jürgen
TI - Global Existence for a Strongly Coupled Cahn-Hilliard System with Viscosity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/10//
PB - Unione Matematica Italiana
VL - 5
IS - 3
SP - 495
EP - 513
AB - An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [3]. Both systems conform to the general theory developed in [5]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $\rho$ and the chemical potential $\mu$. In the system studied in this note, a phase-field equation in $\rho$ fairly more general than in [3] is coupled with a highly nonlinear diffusion equation for $\mu$, in which the diffusivity coefficient is allowed to depend nonlinearly on both variables.
LA - eng
UR - http://eudml.org/doc/290927
ER -

References

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  1. BARBU, V., Nonlinear semigroups and differential equations in Banach spaces, Noordhoff, Leyden, 1976. Zbl0328.47035MR390843
  2. BREZIS, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud.5, North-Holland, Amsterdam, 1973. Zbl0252.47055MR348562
  3. COLLI, P. - GILARDI, G. - PODIO-GUIDUGLI, P. - SPREKELS, J., Well-posedness and long-time behaviour for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870. Zbl1331.74011MR2861103DOI10.1137/110828526
  4. COLLI, P. - GILARDI, G. - PODIO-GUIDUGLI, P. - SPREKELS, J., Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, preprint WIAS-Berlin n. 1713 (2012), 1-28. MR3035431DOI10.1016/j.jde.2013.02.014
  5. PODIO-GUIDUGLI, P., Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118. Zbl1150.74091MR2248166DOI10.1007/s11587-006-0008-8
  6. SIMON, J., Compact sets in the space L p ⁢ ( 0 ; T ; B ) , Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. Zbl0629.46031MR916688DOI10.1007/BF01762360

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